Simplify your Calculations!

Note: This is article #4 in the multi-article series Are you doing it wrong?

In articles 1, 2, and 3, we examined questions where GMAT-specific strategies are more efficient than conventional (i.e., high school math) strategies. In this article, we’ll examine strategies that allow us to bypass time-consuming (and error prone) calculations, regardless of which strategy we’re using to solve the question. 

The strategies we’ll examine are: 

• Approximation
• Fraction properties
• Equivalent fractions 
• Units digit 
• Divisibility Rules

Before we get into the strategies, let’s first address this. . .

The GMAT is not designed to reward human calculators. 

Always remember that the GMAT is designed to measure your quantitative reasoning skills, which includes number sense. That’s why the required calculations in GMAT questions seldom exceed the complexity of calculations like 45 x 19 and 72,000 ÷ 0.24. So, if you find yourself performing painfully long calculations on official GMAT questions, it’s likely that you’re missing an opportunity to reduce the difficulty of your calculations. 

For example, if you catch yourself performing long division on an official GMAT question, there’s a very good chance you’re doing more work than necessary (here’s one of the few exceptions). 

Knowing the test-takers aren’t interested in your ability to perform tedious calculations can sometimes serve as a useful hint. For example, this question from the GMAT Official Guide asks us to evaluate 3.003/2.002, and the answer choices are (A) 1.05, (B) 1.50015, (C) 1.501, (D) 1.5015, and (E) 1.5.  

Since the answer choices are too tightly packed to use any approximation techniques, are we forced to perform long division? Probably not, since the test-makers aren’t interested in this mundane skill. So, before resorting to long division (plan B), let’s give ourselves a few seconds to identify an approach that uses quantitative reasoning. 

In this case, we can factor out 1.001 from the numerator and denominator to get (3)(1.001)/(2)(1.001), which simplifies to 3/2, which equals 1.5. Answer: E. 

Okay, let’s explore some of the ways we can make our calculations easier. 

1. Approximation

Approximation is a huge time saver, plus we’re much less likely to make mistakes when approximating. For example, calculating (0.096)(15.4)(3.7) will take much longer than calculating (0.1)(15)(4), and we’re much less likely to make a mistake on the second calculation.  

Of course, before we start approximating every value, we must understand when it’s okay to approximate, and when it isn’t. In general, the degree to which we can approximate depends on the relative spread of the answer choices. 

For example, in this official GMAT question, we must calculate (385)(100)/7, and the answer choices are (A) 55, (B) 550, (C) 2,695, (D) 5,500 and (E) 26,950

The relative spread among these answer choices is huge. Answer choice B is 900% greater than choice A; choice C is almost 400% greater than choice B; choice D is about 100% greater than choice C; and choice E is almost 400% greater than choice E. This big spread tells us we can be quite aggressive with our approximations.     

So, for example, rather than calculate (385)(100)/7, let’s calculate (350)(100)/7, since 7 divides nicely into 350. 

When we do so, we get: (385)(100)/7 ≈ (350)(100)/7 ≈ 35,000/7 ≈ 5,000, which means the answer must be D, since 5,500 is the only option that’s close to 5,000. . 

The approximation works here because changing 385 to 350 represents a mere 9-ish% decrease, and the smallest relative difference between the answer choices is 100%. 

Alternatively, we could have changed 385 to 420 (the next “nice” multiple of 7) to get: (385)(100)/7 ≈ (420)(100)/7 ≈ 42,000/7 ≈ 6,000, which means the answer is still D, because 5,500 is the only option close to 6,000.   

The GMAT test-makers generously provide tons of opportunities to both reduce our mental workload and save us time. These opportunities, when paired with other GMAT-specific strategies, provide a lot of openings to efficiently identify the correct answer. 

In this official GMAT question, we’re required to convert 102/79 to a percent, and the answer choices are A. 25%, B. 55%, C. 100%, D. 125%, E. 155%

Since the answer choices are moderately spread apart, we can approximate moderately. 

So, we get 102/79 ≈ 100/80 ≈ 5/4 ≈ 1.25 ≈ 125%, which means the correct answer is D. 

It’s important to note that some questions present a dilemma when it comes to approximating. For example, in this question from the Official Guide, we must convert 11,045/8,902 to a percent, and the answer choices are (A) 45%, (B) 125%, (C) 145%, (D) 150%, (E) 225%

While most of the answer choices are somewhat spread apart, the close proximity of choices C and D suggests we shouldn’t approximate. For example, what if our approximation results in an answer such as 148%? If that occurred, we wouldn’t know whether C or D is correct, which means we’d have to perform the actual calculations. 

So, what do we do? 

First off, the question asks, ”Approximately what was the percent increase?” So, it would be a jerk move on the test-makers’ part to tell us to approximate only to end up with some number close to both choices C and D. That said, it’s conceivable this question could require a different calculation-reducing strategy (like the one we’ll examine next) that still works when the answer choice are close together. 

Given all of this, it’s still best to start by approximating, because it’s super fast, which means we won’t have wasted much time if we end having to actually evaluate 11,045/8,902.

So, let’s approximate: 11,045/8,902 ≈ 11,000/9,000 ≈ 11/9 ≈ 1 2/9 ≈ 1.22 ≈ 122%, which means the correct answer must be B. Our “gamble” paid off. 

For more examples of questions (from the Official Guide) where we can use approximation, see this, this, and this (for starters).

Of course, the answer choices don’t always allow us to approximate. In many cases, tightly packed answer choices make approximation prohibitive. This, however, doesn’t mean there aren’t other ways to avoid lengthy calculations. 

Let’s examine some of these strategies, starting with . . . 

2. Fraction properties

We can sometimes avoid tedious calculations by applying one of the following fraction properties: 

If j, k and n are positive numbers then: 

• j/(k + n) < j/k (In other words, k divides more times into j than k+n does)  
• (j+n)/k  > j/k (In other words, k divides into j+n more times than it divides into j)

For example, the fractions 39/14 and 39/15 have the same numerators. Since 14 is less than 15, we know that 14 will divide into 39 more times than 15 will divide into 39, which means 39/14 > 39/15

Similarly, the fractions 41/55 and 42/55 have the same denominators. Since 41 is less than 42, we know 41/55 < 42/55

Let’s see how this property works with this question from the Official Guide:

If 0.497 mark has the value of one dollar, what is the value, to the nearest dollar, of 350 marks?

A. $174

B. $176

C. $524

D. $696

E. $704

We can use equivalent ratios to create the equation 0.497/1 = 350/x, which, upon cross multiplication, simplifies to: x = 350/0.497 

It would be great to simply approximate 350/0.497, but the close proximity of choices A & B and choices D & E would render our approximation efforts futile (unless the correct answer is C). 

Does this mean we must use long division to evaluate 350/0.497?  Absolutely not. 

For this question, we can apply fraction property #1. 

Notice that 350/0.497 is very close in value to the “nicer” fraction 350/0.5, which evaluates to 700. 

Since 0.497 is a little less than 0.5, we know that 0.497 will divide into 350 a little more than 0.5 does. This means 350/0.497 must be a little more than 700, which means E is correct. 

Had the test-makers provided two answer choices that were both a little bit bigger than 700 (e.g.. 704 and 706), then we wouldn’t be able to apply this useful technique. However, instead of requiring us to perform long division, they’re testing our number sense. 

Similarly, this question from the Official Guide requires us to convert 3/12.95 into a percent, and the answer choices are tightly packed: (A) 38%, (B) 31%, (C) 30%, (D) 29% and (E) 23%. 

If we recognize that 3/12 = 25%, then it must be the case that 3/12.95 is less than 25% and, whaddayaknow, only one answer choice is less than 25%. Answer: E. 

Another example of the test-makers creating answer choices that allow us to bypass longer calculations. 

There are so many examples of the test-makers’ intent to reward number sense over tedious calculations. In this question from the Official Guide, we must calculate 57,600/8, and answer choices are (A) $960, (B) $1440, (C) $2880, (D) $4608, and (E) $7200. There are two super-easy ways to identify the correct answer: 

Option 1: Since 56,000/8 = 7,000, we know that 57,600/8 must be greater than 7,000 (E).

Option 2: Since 57,600/10 = 5,760, we know that 57,600/8 must be greater than 5,760 (E).

For more examples of test-makers rewarding students who apply these fraction properties, see this question, this question, and this one (to name a few). 

3 - Equivalent Fractions (plus new notation!)

Another fraction property we can use to simplify our calculations is based on the fact that taking a fraction with a power of 10 (e.g., 100, 1000, 10000, etc.) in the denominator and converting it to a decimal or percent is crazy easy. For example, 17/100 = 0.17 = 17%, and 38/1000 =0.038 = 3.8%. 

Another example where we can save time by applying this concept is this GMAT Official Guide question that requires us to convert 52/325 into a percent. The answer choices are (A) 6%, (B) 12%, (C) 14%, (D) 16% and (E) 20%. 

The efficient approach is to convert 52/325 to an equivalent fraction with 1000 in its denominator. 

Since (325)(3) = 975, we know we must multiply 325 by a number that’s slightly more than 3 to get a denominator of 1000. 

New notation alert! We can use the notation 3+ to denote a number that’s slightly more than 3. 

So, we’ll take 52/325 and create an equivalent fraction by multiplying numerator and denominator by (3+) to get: (52)(3+)/(325)(3+) = 156+/1000 = 0.156+, which converts to a percent that’s slightly greater than 15.6%. So, the answer is D. 

This next question from the Official Guide combines the above strategy with others: 

Q image - 1-over-037 question.png

As you probably already guessed, we’re not required to perform the actual calculations as shown; we’re required to efficiently identify the correct answer (a much different task).

IF the test-makers had actually wanted us to perform these painful calculations, they would have given us tightly-packed answer choices like this: 

A) 0.02725

B) 0.0275

C) 0.02775

D) 0.027775

E) 0.02785

If these were the actual answer choices, we’d have no choice but perform each and every calculation.

Instead, the test-makers gave us answer choices that are well spread apart. Choice B (in the original question) is about 600% greater than A, and C is 9900% greater than B. D is about 30% greater than C, and E is 900% greater than D. So, for the most part, the answers are well spread apart, which means we can replace time-consuming long division with some approximations.

Let’s first evaluate 1/0.03

Since many students struggle to estimate this in their head, I suggest we create an equivalent fraction devoid of decimals. To so, we’ll multiply numerator and denominate by 100 to get 100/3, which is approximately 33. 

We can use the same process to convert 1/0.34 to 100/34, which is approximately 3.

So, the original expression can now be expressed as 1/(33 + 3), which equals 1/36.

Form here, we can quickly estimate the value of 1/36 by converting it into an equivalent fraction with a power of 10 in the denominator. So, let’s convert 1/36 to an equivalent fraction with 100 in the denominator (i.e., 1/36 = ?/100). 

To do so, we must multiply 36 by a number that’s a little bit less than 3, which I’ll write as 3- 

Note: the “-“ represents “a little less than.” It has nothing to do with negative values. 

When we multiply numerator and denominator by 3-, we get: (1)(3-)/(36)(3-) = 3-/100 = 0.03-, which represents a number a little bit less than 0.03.

When we check the answer choices, only one option a little bit less than 0.03. Answer: B.

4. Follow the Units Digits

This next strategy is a popular one. It involves focusing on the units digit for each calculation. For example, this question from the GMAT Official Guide eventually requires us to evaluate 26 + 26² + 26³, and the answer choices are A. 2,951, B. 8,125, C. 15,600, D. 16,302 and E. 18,278. 

Since it’s unlikely the test-makers would create answer choices that would require us to perform all of those calculations, there must be an alternate approach, but what is it?       

If we check the units digits of the five answer choices, we see that they’re all different, which means can just follows the units digits of each calculation.

We know that 26 has units digit 6.   

26² = (26)(26) = ???6 (some number with units digit 6).   

26³ = (26)(26)(26) = ????6 (some number with units digit 6).   

So, 26 + 26² + 26³ = 26 + ??6 + ??6 = ???8, which means the correct answer must be E. 

The test-makers could have made every answer choice have units digit 8, but they’re testing something other than your ability to multiply 26 by itself three times. 

Similarly, this question requires us to evaluate 67 x 12, and the answer choices are (A) 50, (B) 64, (C) 67, (D) 768 and (E) 804

If we follow the units digits, we get 67 x 12 = ???4, which means the correct answer is either B or E. Since the product must be greater than 64, we can be certain the correct answer is E. 

Need more examples from the Official Guide?  Here’s a 650-ish level question where the technique comes in handy. Here’s a 700+ level question, and here’s one that involves a slight twist

5. Divisibility Rules

This last calculation-reducing strategy is based on divisibility rules. For example, one divisibility rule tells us that an integer is divisible by 9 if and only if the sum of its digits is divisible by 9. So, for example, if we encounter a GMAT question where the solution is the product of 9 and 12,432, then we know the sum of the digits of the correct answer must be divisible by 9.  

This strategy doesn’t come into play a lot, but I wanted to include it nonetheless. That said, here’s a tricky question from the official GMAT practice test where I’m certain the test-makers were hell bent on rewarding this kind of number sense: 

Note: This next question is from the official GMAT practice test. So, if you're concerned that you'll see this question later, while taking a practice test, ignore it. 

There are 27 different three-digit integers that can be formed using only the digits 1, 2 and 3. If all 27 of the integers were listed, what would their sum be?

A. 2,704

B. 2,990

C. 5,404

D. 5,444

E. 5,994

Here, we can apply a divisibility rule that says integer N is divisible by 3 if and only if the sum of the digits of N is divisible by 3.

Notice that 1+2+3 = 6, and 6 is divisible by 3.

This means any 3-digit integer consisting of a 1, a 2, and a 3 must be divisible by 3. 

If each of the 27 integers in the sum is divisible by 3, then the entire sum must be divisible by 3. In other words, the correct answer must be divisible by 3, which means the sum of its digits must be divisible by 3.

Let's check each answer choice...

A. 2+7+0+4 = 13, which is not divisible by 3. Eliminate.

B. 2+9+9+0 = 20, which is not divisible by 3. Eliminate.

C. 5+4+0+4 = 13, which is not divisible by 3. Eliminate.

D. 5+4+4+4 = 17, which is not divisible by 3. Eliminate.

E. 5+9+9+4 = 27, which IS divisible by 3.

By the process of elimination, the correct answer is E.

It’s definitely no coincidence the test-makers made exactly ONE answer choice divisible by 3. Also notice that four of the answer choices have units digit 4, which seriously weakens the “following the units digit” strategy, AND they chose closely-packed answer choices to kill any attempts at approximation.   

Here’s another question from the Official Guide you can review, and here’s one my own super-hard questions that also benefit from knowing the divisibility rules.  

Final Words

Over the past 4 articles, we’ve examined a variety of GMAT-specific strategies that can help us identify the correct answer in the most efficient manner possible. 

With all of that in mind, here’s my proposed flowchart for all GMAT Problem Solving questions: 

GMAT Problem Solving Flowchart.png